Standardized Factors and Linear Stochastic Discount Factor Loadings

Summary

The document raises a conceptual question about why a linear stochastic discount factor in an arbitrage pricing framework might use factor risk premia as its coefficients. It presents a return factor model, assumes factor means represent risk premia, and specifies factors with unit variance. It also points to a claim that standardizing observable factors makes the relevant coefficients regression loadings, or betas, of asset payoffs on those factors.

No answer or derivation is included, so the note does not establish when this coefficient interpretation follows or what assumptions are required. In particular, it leaves unresolved the relationship among factor means, factor covariance, regression exposures, and stochastic discount factor pricing restrictions. It is therefore useful as a clearly scoped theoretical question, but readers need a fuller asset pricing treatment before applying the stated linear specification or interpreting its coefficients as prices of risk.

Key ideas

• The note asks how factor risk premia relate to coefficients in a linear stochastic discount factor.
• It frames the question using a factor model with factor means interpreted as risk premia.
• Unit variance standardization is linked to interpreting coefficients as regression loadings.
• The document supplies no derivation, answer, or explicit assumptions that justify this interpretation.

Tags

# linear stochastic discount factor


# linear stochastic discount factor












I have heard some people say something like the following with regards to APT:

Let returns be given by the factor model

$r_t = B_tf_t + e$

with

$E(f_t) = \lambda_t$

Assume that factors are standardized so their variance is 1. Assume a linear sdf like in APT: $m_{t+1} = - \lambda_t f_{t+1}$

While I understand writing the sdf as a linear function of factors, I'm not sure why the coefficients of factors are price of risk to factor itself?

Is there a justification for this?

I found this note which seems to talk about it, but I'm still not clear on this. Specifically, page 7 top LHS states:

"When the entries of z are standardized to a have conditional variances equal to unity, the entries of λ become the conditional regression coefficients, the “betas” of the asset payoffs onto the alternative observable factors"

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.